As For the Other Shapes…

Shape \(A_n\) as described on the previous page is just one of many root polytopes! For our research purposes, we will only be considering the shapes \(A_n, B_n, C_n,\) \(and\) \(D_n\), since these shapes can take on any number of n dimensions.

Root Polytopes \(B_n, C_n\) and \(D_n\)

The root polytopes are defined as follows

\(\color{blue}{\text{Proposition:}}\) The root polytope \(B_n, C_n\) and \(D_n\) are balanced. The following polytopes are what \(B_3\), \(C_3\) and \(D_3\) look like.

Spectra of \(B_n\), \(C_n\) and \(D_n\) and Future Work to Come…

We are only just starting to develop an understanding of the patterns of these root polytopes. The following patterns listed for the eigenvalues of polytopes \(B_n\), \(C_n\) and \(D_n\) are based off of our computations, and plans to complete proofs for each of these shapes is in the works!

The \(\color{green}{\text{spectrum}}\) of \(TL(B_n)\) is: \[ \frac{3\pm \sqrt{16n^2-40n+33}}{2}, \underbrace{0}_n, \underbrace{ \frac{2n+3\pm\sqrt{4n^2-20n+33}}{2}}_{n-1},\underbrace{2n-3}_{\frac{n(n-1)}{2}}, \underbrace{2n-1}_{n(n-1)}, \underbrace{2n+1}_{\frac{n(n-3)}{2}}\]

The \(\color{green}{\text{spectrum}}\) of \(TL(C_n)\) is \[\frac{(5-n)\pm \sqrt{9n^2-26n+33}}{2}, \underbrace{0}_n,\underbrace{ \frac{(5+n)\pm\sqrt{n^2-6n+33}}{2}}_{n-1}, \]\[\underbrace{n+1}_{n-1}, \underbrace{2(n-1)}_{\frac{n(n-1)}{2}}, \underbrace{2n}_{n(n-2)} ,\underbrace{2(n+1)}_{\frac{n(n-3)}{2}}\]

The \(\color{green}{\text{spectrum}}\) of \(TL(D_n)\) is \[2(2-n), \underbrace{4}_{n-1},\underbrace{0}_{n},\underbrace{2n}_{\frac{n(n-3)}{2}}, \underbrace{2(n-2)}_{\frac{n(n-1)}{2}},\underbrace{2(n-1)}_{n(n-2)}\]

References

Farhad Babaee and June Huh, A Tropical Approach to a Generalized Hodge Conjecture for Positive Currents, Duke Math. J. 166 (2017), no. 14, 2749–2813. MR 3707289

Anna Schindler, Algebraic and Combinatorial Aspects of Two Symmetric Polytopes, Master’s thesis, San Francisco State University, 2017.